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On solitons, compactons, and Lagrange maps. (English) Zbl 1059.35524

Summary: Two local conservation laws of the \(K(m,n)\) equation \(u_ t\pm(u^ m)_ x+(u^ n)_ {xxx}=0\) are used to define two Lagrange-type transformations into mass and momentum space. These mappings help to identify new integrable cases \((K(-1,-2)\), \(K(-2,-2)\), \(K(\frac32,-\frac12))\), transform conventional solitary waves into compactons – solitary waves on a compactum – and relate certain soliton-carrying systems with compacton-carrying systems. Integrable equations are transformed into new integrable equations, and interaction of \(N\)-solitons of the \(m\)-KdV \((m=3, n=1)\) is thus projected into an interaction in a compact domain from which \(N\) ordered stationary compactons emerge. The interaction of traveling compactons is the image of super-imposed equilibria of the corresponding soliton equation. For \(m=n+2\), the potential form of the \(K(m,n)\) equation may also be cast into a conserved form and thus transformed, yielding generalized Dym and Wadati equations and two new integrable cases. It is shown that \(r_ t+(1-r^ 2)^ {3/2}(r_ {xx}+r)_ x=0\) is integrable and supports compact kinks.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
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References:

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